zbMATH — the first resource for mathematics

On time-harmonic Maxwell equations with nonhomogeneous conductivities: Solvability and FE-approximation. (English) Zbl 0696.65085
The authors analyze the solvability of the 3D problem: $$rot H=E$$ in $$\Omega,$$ $$rot E=-i^{\omega \mu}H$$ in $$\Omega$$ and $$nx E=nx \tilde E$$ on $$\partial\Omega,$$ where E, H are complex-valued vector functions independent of time, $$\epsilon$$ is the electric dielectricity, $$\mu$$ the magnetic permeability, $$\sigma$$ is the $$3\times 3$$ matrix of electric conductivities, $$\tilde E$$ a given vector function. An existence theorem for the solution is proved, and an error estimation for a finite element approximation of the 3D problem is presented. Some particular results for the 2D problem are obtained. Two numerical experiments are presented.
Reviewer: I.Grosu

MSC:
 65Z05 Applications to the sciences 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 78A25 Electromagnetic theory (general) 35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text:
References:
 [1] V. Bezvoda K. Segeth: Mathematical modeling in electromagnetic prospecting methods. Charles Univ., Prague, 1982. [2] E. B. Byhovskiy: Solution of a mixed problem for the system of Maxwell equations in case of ideally conductive boundary. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 12 (1957), 50-66. [3] V. Červ K. Segeth: A comparison of the accuracy of the finite-difference solution to boundary-value problems for the Helmholtz equation obtained by direct and iterative methods. Apl. Mat. 27 (1982), 375-390. · Zbl 0511.65074 [4] P. G. Ciarlet: The finite element method for elliptic problems. North-Holland, Amsterdam, New York, Oxford, 1978. · Zbl 0383.65058 [5] D. Colton L. Päivärinta: Far field patterns and the inverse scattering problem for electromagnetic waves in an inhomogeneous medium. Math. Proc. Cambridge Philos. Soc. 103 (1988), 561-575. · Zbl 0669.35109 [6] G. Duvaut J. L. Lions: Inequalities in mechanics and physics. Springer-Verlag, Berlin, 1976. · Zbl 0331.35002 [7] V. Girault P. A. Raviart: Finite element methods for Navier-Stokes equations. Springer- Verlag, Berlin, Heidelberg, New York, Tokyo, 1986. · Zbl 0413.65081 [8] P. Grisvard: Boundary value problems in nonsmooth domains. Lecture Notes 19, Univ. of Maryland, Dep. of Math., College Park, 1980. [9] G. Heindl: Interpolation and approximation by piecewise quadratic $$C^1$$-functions of two variables. in Multivariate approximation theory by W. Schempp and K. Zeller. ISNM vol. 51, Birkhäuser, Basel, 1979, pp. 146-161. [10] E. Hewitt K. Stromberg: Real and abstract analysis. Springer-Verlag, New York, Heidelberg, Berlin, 1975. [11] J. Kadlec: On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of convex open set. Czechoslovak Math. J. 14 (1964), 386-393. · Zbl 0166.37703 [12] F. Kikuchi: An isomorphic property of two Hubert spaces appearing in electromagnetism. Japan J. Appl. Math. 3 (1986), 53-58. · Zbl 0613.46040 [13] M. Křížek P. Neittaanmäki: Solvability of a first order system in three-dimensional nonsmooth domains. Apl. Mat. 30 (1985), 307-315. · Zbl 0593.35073 [14] S. Mareš, al.: Introduction to applied geophysics. (Czech). SNTL, Prague, 1979. [15] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. · Zbl 1225.35003 [16] P. Neittaanmäki R. Picard: Error estimates for the finite element approximation to a Maxwell-type boundary value problem. Numer. Funct. Anal. Optim. 2 (1980), 267-285. · Zbl 0469.65079 [17] J. Saranen: On an inequality of Friedrichs. Math. Scand. 51 (1982), 310-322. · Zbl 0524.35100 [18] R. S. Varga: Matrix iterative analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1962. · Zbl 0133.08602 [19] V. V. Voevodin, Ju. A. Kuzněcov: Matrices and computation. (Russian). Nauka, Moscow, 1984. [20] Ch. Weber: A local compactness theorem for Maxwell’s equations. Math. Methods Appl. Sci. 2 (1980), 12-25. · Zbl 0432.35032 [21] Ch. Weber: Regularity theorems for Maxwell’s equations. Math. Methods Appl. Sci. 3 (1981), 523-536. · Zbl 0477.35020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.